Equivariant Convolution: 
To canonicalize for rotation, we leverage equivariance properties of spherical harmonic functions to obtain rotation-equivariant learnable features on the field [1]}, [2]}.
Type-\(\ell \)  spherical harmonic functions (\(Y^{\ell }\) ) are degree-\(\ell \)  polynomials of points on the sphere that are equivariant to \(SO(2\ell + 1)\) .
The densities sampled from NeRF reconstructions are often noisy and contain outliers.
To promote our method to learn only from occupied regions, we weigh each equivariant feature type by its density.
This weighing has no effect on the equivariance of features (see supplementary document for proof).
We have an option to weigh equivariant features with the local average of the density to smoothen the signal or to weigh directly by the density. We found weighing by density working better than taking the local average density (see sec:expt).
We learn equivariant features of type \(\ell \)  as:
\(& {}^{R}s^{\ell } := f_{w}({}^{R}x) \left(W^\ell (\text{EQConv}^\ell ({}^{R}x)) + \delta _{0\ell }b\right), \\&f_{w}({}^{R}x) :={\left\lbrace \begin{array}{ll}\text{mean}_{r} (\mathbf {\sigma }_D({}^{R}x)), & or\\\mathbf {\sigma }_D({}^{R}x),\end{array}\right.},\) 
